1. Field of the Invention
The present invention relates to computed tomographic (CT) imaging, and in particular to CT imaging utilizing up-sampling with shift.
2. Discussion of the Background
The main components of a computed tomography (CT) system are an x-ray tube and x-ray detector. The x-ray tube produces x-ray photons directed towards a scanned object or patient. When x-ray photons penetrate through the object or patient they lose intensity (attenuate) and are measured by the x-ray detector, located on the opposite side of the object or patient. The x-ray source and detector are typically attached to a ring-like base, called a gantry. The gantry rotates around the scanned object or patient, allowing collecting data at a multitude of view angles. At each view angle one x-ray projection is collected and transferred through the digital acquisition system (DAS) to the data processor unit and reconstruction unit. Each projection is given as a set of measurements by a two-dimensional detector array of the intensity of the x-rays emanating from the x-ray source. Such data is called “cone beam” data.
In a two-dimensional detector where each detector element is indexed by k, k=1 . . . N×M, where N is the number of detector rows, and M is the number of elements per detector row, the intensity of the x-ray photon beam (ray) at the detector element k, attenuated by an object or patient, is given by:Ik=Ik0 exp(−∫μ(x)dx)  (1)where:
γ(x) is the attenuation function sought to be reconstructed.
Ik0 is the beam intensity before attenuation by γ(x), as produced by the x-ray tube and after penetrating through the x-ray filter (for example, wedge or bowtie); and
∫μ(x)dx is the line integral of γ(x) along the line l.
Mathematically, γ(x) can be reconstructed given a set of line integrals corresponding to a plurality of lines l. Therefore, measured intensity data needs to be converted into line integrals first:∫μ(x)dx=ln(Ik0)−ln(Ik)  (2)X-ray tomographic reconstruction consists of the main steps of data acquisition, data processing and data reconstruction. In data acquisition, x-ray intensity data is collected at each detector element and each predefined angular view position. This is done within the rotating part of the gantry. Detectors measure incident x-ray flux and convert it into an electric signal. There are two main types of detectors: energy (charge) integrating and photon counting. The electrical signal is transferred from the rotating part of the gantry to the stationary part though the slipring. During this step data may be compressed.
In data processing, the data is converted from x-ray intensity measurements to the signal corresponding to line integrals according to equation (2). Also, various corrections steps may be applied to (1) reduce effects of undesired physical phenomena, such as scatter, x-ray beam hardening, (2) compensate non-uniform response function of each detector element, and (3) reduce noise.
Depending on the algorithm, data reconstruction may contain all or some of the following processing steps                Cosine (fan angle, cone angle) weighting (can be ×cos, or 1/cos)        Data differentiation: This can be performed with respect to fan angle, cone angle, projection angle, source trajectory coordinate, vertical detector coordinate, horizontal detector coordinate, or any combination of those.        Data redundancy weighting. Data is multiplied by the weight function W, which may be a function of fan angle, cone angle, projection angle, source trajectory coordinate, vertical detector coordinate, horizontal detector coordinate, or any combination of those.        Convolution (filtering). This step utilizes a convolution kernel. Some algorithms use ramp-based kernel (H(w)=|w|), some use Hilbert-based kernel (h(t)=1/t, h(t)=1/sin(t), H(w)=i sign(w)). Kernels can be adjusted to the fan beam geometry, scaled, modulated, apodised, modified, or any combination of those.        Backprojection. In this step data is projected back in the image domain. Usually, backprojected data is weighted by a distance factor. The distance factor is inversely proportional to the distance L from the x-ray source position to the reconstructed pixel, and can be proportional to 1/L or 1/L2. Also, some additional data redundancy weighting can be applied on the pixel-by-pixel basis. Also, usually the backprojection step includes obtaining data values corresponding to the ray through the reconstructed pixel by either data interpolation or data extrapolation. This process can be done in a numerous variety of ways.The order in which the above steps are applied depends on a specific reconstruction algorithm.        
It is possible to translate the table, on which the scanned object or patient lies, during the scan, so that the object or patient is translated through the gantry. In this case x-ray source describes helical trajectory, relative to the scanned object or patient. This is called “helical cone beam” scanning.
Because of the finite detector element size, projection data corresponds to the band-limited (smoothed) version, γ0(x), of the real reconstructed function γ(x). Therefore, at best, γ0(x) can be reconstructed, not γ(x). However, according to the Nyquist criterion, to reconstruct γ0(x) two samples per beam width are needed. Therefore, if the scanned function γ(x) is sufficiently sharp it violates the Nyquist criterion which causes aliasing artifacts. With helical scans insufficient sampling along the z-axis causes an artifact pattern with interleaving dark and light blades (leafs), known as the “windmill artifact” or “helical artifact”.